Go to previous page Go up Go to next page

4.7 The Weyl group

The Weyl group 𝔚 [𝔤] of a Kac–Moody algebra 𝔤 is a discrete group of transformations acting on 𝔥∗. It is defined as follows. One associates a “fundamental Weyl reflection” ri ∈ 𝔚 [𝔤] to each simple root through the formula
(λ|αi)- ri(λ ) = λ − 2(αi|αi )αi. (4.55 )
The Weyl group is just the group generated by the fundamental Weyl reflections. In particular,
ri(αj ) = αj − Aijαi (no summation on i). (4.56 )

The Weyl group enjoys a number of interesting properties [116Jump To The Next Citation Point]:


Table 13: Cartan integers and Coxeter exponents.
AijAji mij
0 2
1 3
2 4
3 6
≥ 4 ∞

This close relationship between Coxeter groups and Kac–Moody algebras is the reason for denoting both with the same notation (for instance, An denotes at the same time the Coxeter group with Coxeter graph of type An and the Kac–Moody algebra with Dynkin diagram An).

Note that different Kac–Moody algebras may have the same Weyl group. This is in fact already true for finite-dimensional Lie algebras, where dual algebras (obtained by reversing the arrows in the Dynkin diagram) have the same Weyl group. This property can be seen from the fact that the Coxeter exponents are related to the duality-invariant product AijAji. But, on top of this, one sees that whenever the product AijAji exceeds four, which occurs only in the infinite-dimensional case, the Coxeter exponent m ij is equal to infinity, independently of the exact value of AijAji. Information is thus clearly lost. For example, the Cartan matrices

( ) ( ) 2 − 2 − 2 2 − 9 − 8 ( − 2 2 − 2) , ( − 4 2 − 5) (4.57 ) − 2 − 2 2 − 3 − 7 2
lead to the same Weyl group, even though the corresponding Kac–Moody algebras are not isomorphic or even dual to each other.

Because the Weyl groups are (crystallographic) Coxeter groups, we can use the theory of Coxeter groups to analyze them. In the Kac–Moody context, the fundamental region is called “the fundamental Weyl chamber”.

We also note that by (standard vector space) duality, one can define the action of the Weyl group in the Cartan subalgebra 𝔥, such that

⟨γ,r∨(h )⟩ = ⟨r (γ),h⟩ for γ ∈ 𝔥⋆ and h ∈ 𝔥. (4.58 ) i i
One has using Equations (4.30View Equation, 4.32View Equation, 4.33View Equation, 4.35View Equation),
r∨(h) = h − ⟨α ,h ⟩h = h − 2 (h|hi)h . (4.59 ) i i i (hi|hi) i

Finally, we leave it to the reader to verify that when the products AijAji are all ≤ 4, then the geometric action of the Coxeter group considered in Section 3.2.4 and the geometric action of the Weyl group considered here coincide. The (real) roots and the fundamental weights differ only in the normalization and, once this is taken into account, the metrics coincide. This is not the case when some products AijAji exceed 4. It should be also pointed out that the imaginary roots of the Kac–Moody algebras do not have immediate analogs on the Coxeter side.

Examples


  Go to previous page Go up Go to next page